The penalty was introduced by a Labour Government in 1990 and has been kept in a rewrite of the Social Security Act now back to Parliament from the social services select committee yesterday.
But the College of Midwives, Women's Refuge, Child Poverty Action Group and Auckland Action Against Poverty (AAAP) are campaigning to scrap the penalty.
The penalty takes $22 a week off the benefit for each child whose father is not named, plus an extra $6 for each family if a father is still not named after 13 weeks.
At the end of March 10,848 parents were being docked for one child, 2189 for two children, 476 for three children, 82 for four children and 21 for five or more children - affecting a total of at least 17,087 children.
More than half those affected (52.8 per cent) were Maori.
The Auckland woman, whom the Herald agreed to call "Jane", said she had "nothing left" to buy nappies for her daughter while the $28 was being deducted. She had to rely on family support.
"I was asking for help for nappies and that," she said.
She used up all her food grant entitlements and had to ask a local church for food parcels three or four times.
She said Work and Income insisted that she name her daughter's father, but she couldn't.
"I don't know who the father is," she said. "They said I have to put the name of the father and tell them to contact whoever I was with at the time, but I wasn't with anybody at the time."
The law allows for exemptions in cases of violence or where there is "insufficient evidence available to establish who is in law the other parent".
Jane said she was not told about the exemption until she met an AAAP advocate. She was referred to a community law centre where a lawyer wrote to Work and Income explaining why she couldn't name her child's father.
Work and Income then cancelled the $28 penalty and a month ago paid her back pay for the full amount she had lost over two years.
Labour MP Carmel Sepuloni said there was "general agreement" in the Labour caucus that the penalty should be scrapped.
"We are a different party in 2016 than we were in 1990," she said.
LISTEN: Carmel Sepuloni: Deadbeat dads hurting children through WINZ law
NZ First MP Darroch Ball said his caucus was unanimously against the penalty.
Maori Party co-leader Marama Fox said the penalty was "an archaic attack on the poor and most vulnerable children of this nation".
But Act MP David Seymour said the penalty should stay to "hold fathers accountable" for supporting their children.
United Future MP Peter Dunne said he would need to see more information on whether the policy was "still fit for purpose within modern New Zealand".
"In the event that the policy is proving to be overly punitive with no tangible benefit to the lives of the children in question, we would be expecting changes to be made."